Table of Contents

Symmetric group

A symmetric group is any member of a certain sequence $(S_N)_{N\in \N}$ of classical orthogonal matrix groups.

Definition

For every $N\in \N$ the symmetric group for dimension $N$ is the subgroup of the general linear group $\mathrm{GL}(N,\C)$ given by all permutation $N\times N$-matrices, i.e., the set

$$S_N\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, \forall_{i,j=1}^N: u_{i,j}\in\{0,1\},\,{\textstyle\sum_{\ell=1}^N} u_{i,\ell}={\textstyle\sum_{k=1}^N} u_{k,j}=1\},$$

where $u=(u_{i,j})_{i,j=1}^N$.

Basic properties

The symmetric groups $(S_N)_{N\in \N}$ are an easy family of compact matrix quantum groups, i.e., the intertwiner spaces of their corepresentation categories are induced by a category of partitions. In fact, the category of all partitions induces the corepresentation categories of $(S_N)_{N\in \N}$. It is canonically generated by $\{\crosspart,\fourpart, \singleton\}$.

Representation theory

Cohomology

References